Simple implementation to find the maximum flow through a flow network (no Capacity Scaling)

"0/10" means an edge with capacity 10 and 0 flow assigned

• Residual graph
  Directed graph showing how much of the flow assignments can be undone
  Edge e=(u,v) shows how much flow is left to be assigned to that edge
  Edge e=(v,u) is how much flow already been assigned (how much ca be undone)
• Augmentation Path
  An arbitrary path from S to T found in residual graph
• Bottleneck
  Minimum capacity from the path found in the residual graph (how much flow is assigned after each path is found)

• vertexCount must be hard coded for different graphs
• Code deals directly with integer array indexes. Index 0 is S (source) and index vertexCount-1 is T (sink) (-1 since arrays count from 0)
• arrayIndexStringEquivalents holds string equivalents of the index. So vertices can have humanly readable IDs like S & T, and don't have to be sequential integers
  • arrayIndexStringEquivalents[1] is 2 because the vertex called "2" is the in position 1 in the array indexes
• Graphs are represented as adjacency matrices
  • Directed Graph so order matters
  • graphMatrix[i][j] represents the capacity of an edge from i to j
    • 0 if no edge exists
    • No vertex has an edge to itself. There should always be a 0 in graphMatrix[x][x]
      • graphMatrix[0][0] is 0 since S has no edges to itself
      • Same for all vertices
      • graphMatrix[1][1] is 0 since the vertex in position 1 (vertex "2") has no edges to itself, etc.
    • Row vertexCount-1 is always all 0's since T (the sink) has NO outgoing edges {0, 0, 0, 0, 0, 0, 0, 0}

• Constructor takes in an array of strings to have human readable vertices instead of ints
• maxFlow() finds the maximum flow through the network
  • Creates a residual graph & leaves the original graph unchanged
• Uses Breadth First Search bfs() to find augmentation paths
  • Returns true/false if a path from S to T was found
  • Updates parent[] array so the actual path can be found
• maxFlow() outputs each augmentation path found & the amount of flow added to total flow (bottleneck)